The injective edge coloring for a class of graphs with maximum degree 5

A [Formula: see text]-injective-edge coloring of a graph [Formula: see text] is an edge coloring [Formula: see text], such that [Formula: see text] for any three consecutive edges [Formula: see text], [Formula: see text] and [Formula: see text] in [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are considered consecutive if they form a path or a cycle of length 3. The minimum integer [Formula: see text] for which [Formula: see text] has a [Formula: see text]-injective-edge coloring is called the injective chromatic index of [Formula: see text], denoted by [Formula: see text]. In this paper, we prove that every graph [Formula: see text] with maximum degree 5 has [Formula: see text] if [Formula: see text]. This result improves upon the previous work by Zhu et al., who showed that [Formula: see text] under the same conditions [Injective edge chromatic number of sparse graphs, Appl. Math. Comput. 473 (2024) 128668].